Ramsey simplicity of random graphs

TL;DR

This paper investigates the conditions under which random graphs are $q$-Ramsey simple, extending previous results to a broader range of probabilities and Ramsey parameters, revealing new behaviors in sparse regimes.

Contribution

It characterizes when $G(n,p)$ random graphs are $q$-Ramsey simple for various $p$ and $q$, broadening understanding of Ramsey properties in sparse random graphs.

Findings

$G(n,p)$ is almost surely $2$-Ramsey simple when $rac{ ext{log } n}{n} ext{ and } p ext{ are in certain ranges}$

The paper uncovers new behaviors for $p$ in the range $n^{-2/3} ext{ to } n^{-1/2}$

It provides a comprehensive analysis of $q$-Ramsey simplicity across different regimes of $p$ and $q$.

Abstract

A graph $G$ is $q$-Ramsey for another graph $H$ if in any $q$-edge-colouring of $G$ there is a monochromatic copy of $H$, and the classic Ramsey problem asks for the minimum number of vertices in such a graph. This was broadened in the seminal work of Burr, Erd\H{o}s, and Lov\'asz to the investigation of other extremal parameters of Ramsey graphs, including the minimum degree. It is not hard to see that if $G$ is minimally $q$-Ramsey for $H$ we must have $\delta(G) \ge q(\delta(H) - 1) + 1$, and we say that a graph $H$ is $q$-Ramsey simple if this bound can be attained. Grinshpun showed that this is typical of rather sparse graphs, proving that the random graph $G(n,p)$ is almost surely $2$-Ramsey simple when $\frac{\log n}{n} \ll p \ll n^{-2/3}$. In this paper, we explore this question further, asking for which pairs $p = p(n)$ and $q = q(n,p)$ we can expect $G(n,p)$ to be $q$-Ramsey simple. We resolve the problem for a wide range of values of $p$ and $q$; in particular, we uncover some interesting behaviour when $n^{-2/3} \ll p \ll n^{-1/2}$.